France



Very recently magnetic stripe domains, characterized by alternating up and down out-of-plane orientation of the magnetization, have received great interest due to the possibility to use stripe patterns to manipulate spin-wave (SW) propagation as in artificial magnonic crystals [1]. In this work, we demonstrate the control of the SW propagation by using reconfigurable regular stripe-pattern domain structure in the hybrid system. The investigated system consists of 64-nm-thick NdCo layer and 10-nm-thick NiFe layer, coupled through an Al layer of different thicknesses. Magnetic force microscopy (MFM) measurements show that, due to the perpendicular magnetic anisotropy of the NdCo film, the system develops stripe domains aligned with the last in-plane saturation direction [2]. The domain pattern is found to have a period of about 140 nm, which is almost independent on the thickness of the Al layer.&lt;br/&gt;The magnetization reversal of the trilayer system was investigated by vibrating sample magnetometer, showing that the hysteresis loop is characterized by a two-step process, due to the different coercivity of the NiFe and NdCo films. Detailed analysis of the hysteresis loops along with micromagnetic simulations indicates that the stray magnetic field coming from the NdCo layer induces a regular domain structure also in the NiFe layer, which is tuned by the thickness of Al spacer. In addition, upon reversing the applied magnetic field, an antiparallel state, characterized by an antiparallel alignment of the magnetization component parallel to the domain axis in the NdCo and NiFe stripes, is formed. Then, Brillouin light scattering spectroscopy has been used to measure the spectra of the SWs propagating in the direction perpendicular to stripe domains for the parallel and the antiparallel state. For both configurations, the dispersion relation shows a strongly nonreciprocal mode (Figs. 1-2). However, in the parallel state SWs propagating with positive and negative wavevector are both characterized by a positive dispersion, while in the reversed state SWs propagating with negative wavevector show a negative dispersion. The above experimental results have been satisfactorily reproduced by numerical simulations. The latter show that the detected SW mode is mainly localized in the NiFe layer and its frequency nonreciprocity can be ascribed to the static magnetization configuration as well as to the interaction with the NdCo induced by the SWs via the dynamic stray field.

**References:**

[1] C. Banerjee et al., Phys. Rev. B 96, 024421 (2017)&lt;br/&gt;[2] D. Markó et al., Appl. Phys. Lett. 115, 082401 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1f952ef7a0b2bb8cf6cafa5252632c82.jpg)
Fig. 1. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the parallel state. Empty and full dots mark the results of Brillouin light scattering measurements and simulations, respectively. Size and color intensity of the numerical results indicate the predicted intensity in the measurements.

![](https://assets.underline.io/uploads/markdown_image/1/image/dbd287cdf366b5549a79bdda3afc82b8.jpg)
Fig. 2. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the antiparallel state.

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 Reconfigurable spin-wave propagation in magnetic stripe domains in hybrid system

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Very recently magnetic stripe domains, characterized by alternating up and down out-of-plane orientation of the magnetization, have received great interest due to the possibility to use stripe patterns to manipulate spin-wave (SW) propagation as in artificial magnonic crystals [1]. In this work, we demonstrate the control of the SW propagation by using reconfigurable regular stripe-pattern domain structure in the hybrid system. The investigated system consists of 64-nm-thick NdCo layer and 10-nm-thick NiFe layer, coupled through an Al layer of different thicknesses. Magnetic force microscopy (MFM) measurements show that, due to the perpendicular magnetic anisotropy of the NdCo film, the system develops stripe domains aligned with the last in-plane saturation direction [2]. The domain pattern is found to have a period of about 140 nm, which is almost independent on the thickness of the Al layer. The magnetization reversal of the trilayer system was investigated by vibrating sample magnetometer, showing that the hysteresis loop is characterized by a two-step process, due to the different coercivity of the NiFe and NdCo films. Detailed analysis of the hysteresis loops along with micromagnetic simulations indicates that the stray magnetic field coming from the NdCo layer induces a regular domain structure also in the NiFe layer, which is tuned by the thickness of Al spacer. In addition, upon reversing the applied magnetic field, an antiparallel state, characterized by an antiparallel alignment of the magnetization component parallel to the domain axis in the NdCo and NiFe stripes, is formed. Then, Brillouin light scattering spectroscopy has been used to measure the spectra of the SWs propagating in the direction perpendicular to stripe domains for the parallel and the antiparallel state. For both configurations, the dispersion relation shows a strongly nonreciprocal mode (Figs. 1-2). However, in the parallel state SWs propagating with positive and negative wavevector are both characterized by a positive dispersion, while in the reversed state SWs propagating with negative wavevector show a negative dispersion. The above experimental results have been satisfactorily reproduced by numerical simulations. The latter show that the detected SW mode is mainly localized in the NiFe layer and its frequency nonreciprocity can be ascribed to the static magnetization configuration as well as to the interaction with the NdCo induced by the SWs via the dynamic stray field.

**References:**

[1] C. Banerjee et al., Phys. Rev. B 96, 024421 (2017) [2] D. Markó et al., Appl. Phys. Lett. 115, 082401 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1f952ef7a0b2bb8cf6cafa5252632c82.jpg)
Fig. 1. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the parallel state. Empty and full dots mark the results of Brillouin light scattering measurements and simulations, respectively. Size and color intensity of the numerical results indicate the predicted intensity in the measurements.

![](https://assets.underline.io/uploads/markdown_image/1/image/dbd287cdf366b5549a79bdda3afc82b8.jpg)
Fig. 2. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the antiparallel state.

technical paper

Nowadays a lot of effort is put into the creation of a practical quantum computer. For this ambitious aim different research fields propose and investigate new materials and technologies. One of such promising systems, a magnon Bose-Einstein condensate (BEC)in magnetic crystals, may allow the creation of magnon-BEC-based qubit at room temperature. A prerequisite for such a qubit is the appearance of spontaneous coherence in magnon BEC. Also, the presence of coherence of the magnon Bose-Einstein condensate is essential for understanding such fundamental properties of it as the possibility of a dissipative magnon supercurrent [1], excitation of Josephson oscillations [2], and propagation of Bogoliubov waves [3]. In most of the previous studies in this field, the magnon BEC is investigated by means of Brillouin light scattering (BLS) spectroscopy [4,5] delivering information about the spectral density of a magnon gas. Unfortunately, due to the limited frequency resolution of the optical Fabry-Pérot interferometers used in BLS facilities, the conventional BLS technique does not allow one to prove the coherence of a magnon BEC directly. The insufficient frequency resolution makes it impossible to separate the relaxation dynamics of condensed and thermal magnons. Moreover, the possible outflow of the condensate from a spatially localized probing light spot complicates the interpretation of the obtained experimental results (see [6] and the corresponding discussion in [7]). For all these reasons only indirect indications of the coherency, obtained on the magnon BEC in magnetic crystals, could be provided yet. Up to this day, a decisive proof of the full coherence magnon BEC remained elusive. In our work, we report a first direct experimental observation of the evolution of magnon BEC towards full coherence, whose degree is limited only by the natural processes of the magnon relaxation into the crystal lattice. To achieve this, we use a novel concept that enhances the coupling of the parametrically-driven magnon system with the probing electromagnetic field. The obtained electric signal was investigated time-resolved by a spectrum analyzer to reach the required frequency resolution necessary for the proper display of the magnon BEC state. A pulsed electromagnetic parametric pumping was used to inject magnons, increase, thus the chemical potential of the magnon gas, and trigger the BEC formation process. It is found that the radiation spectrum picked up after the terminization of a pump pulse is narrowed by two orders of magnitude, compared to the broad-band spectrum obtained during the process of parametric pumping. The resulting spectrum width is only limited by the magnon relaxation frequency. We study this spectrum evolution in many details and identify the conditions for BEC creation. We believe that this direct demonstration of the coherence of the magnon BEC brings the implementation of the room temperature BEC-based computing closer. This research was funded by the European Research Council within the Advanced Grant No. 694709 “SuperMagnonics” and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the Transregional Collaborative Research Center – TRR 173 – 268565370 “Spin+X” (project B01). The authors are grateful to G. A. Melkov and H. Yu. MusiienkoShmarova for the fruitful discussions.

**References:**

[1] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Supercurrent in a room-temperature Bose-Einstein magnon condensate, Nat. Phys. 12, 1027 (2016). [2] A. J. E. Kreil, A. Pomyalov, V. S. L’vov, H. Yu. Musiienko-Shmarova, G. A. Melkov, A. A. Serga, and B. Hillebrands, Josephson oscillations in a room-temperature Bose-Einstein magnon condensate, arXiv:1911.07802. [3] D. A. Bozhko, A. J. E. Kreil, H. Yu. Musiienko-Shmarova, A. A. Serga, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Bogoliubov waves and distant transport of magnon condensate at room temperature, Nat. Commun. 10, 2460 (2019). [4] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping, Nature 443, 430–433 (2006). [5] A. A. Serga, V. S. Tiberkevich, C. W. Sandweg, V. I. Vasyuchka, D. A. Bozhko, A. V. Chumak, T. Neumann, B. Obry, G. A. Melkov, A. N. Slavin, and B. Hillebrands, Bose-Einstein condensation in an ultra-hot gas of pumped magnons, Nat. Commun. 5, 3452 (2014). [6] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, and A. N. Slavin, Observation of spontaneous coherence in Bose-Einstein condensate of magnons, Phys. Rev. Lett. 100, 047205 (2008). [7] A. J. E. Kreil, H. Y. Musiienko-Shmarova, S. Eggert, A. A. Serga, B. Hillebrands, D. A. Bozhko, A. Pomyalov, and V. S. L’vov, Tunable space-time crystal in room-temperature magnetodielectrics, Phys. Rev. B 100, 020406(R) (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/72224fb139fe95447f22f54475f3939d.jpg)
Fig. 1 Sketch of the experimental setup. The sample holder section shows an open waveguide with an internally mounteddielectric resonator. A bulk YIG sample is placed in an opening of the resonator together with a large detection loop to detect the radiated signals from the whole sample volume.

![](https://assets.underline.io/uploads/markdown_image/1/image/84026c2fe3ed134957adcd953bd21b38.jpg)
Fig. 2 Radiated signal of the magnon BEC measured for different times during the free decay of the system. The signals demonstrate a lorentz shape with a FWHM bandwidth of approx. 1 MHz. In normalized view it is visible that the BEC signal demonstrates an unchanged singel-state profile for a long observation period.

 Evolution of room-temperature magnon gas toward coherent Bose-Einstein condensate

In thin-films-based experiments, spin waves (SWs) are often investigated in confined geometries or as plane waves in full films. There are exceptions, such as SW channeling along domain walls [1], which are appealing in terms of SW steering. However, this approach relies on the preparation of a specific magnetic texture and transferring the SW out of it, back into the domains, seems challenging. An alternative route towards the steering of strong, channeled SWs exists, namely caustic SW beams (CSWBs). Caustic phenomena in anisotropic wave propagation correspond to sharply-defined lines/surfaces with large intensities [2]. They have been investigated in a variety of fields [2], and can be expected in micromagnetism. In thin films, caustics can arise due to the marked anisotropy of the SW dispersion relation (DR), illustrated in Fig.1. From such a plot, the direction θv of the group velocity vg=▽kω can be determined from the local normal to the curve. It can be seen in Fig.1 that a portion of the red curve (at 8.5 GHz) is almost straight, leading to a stationary θv around a value θvc for this range of wavefront angles φ. Thus, an excitation of SWs with a broad wave vector spectrum would lead to strong emission in the direction θvc. The corresponding anisotropic enhancement defines a CSWB. Such beams have so far been studied notably upon scattering of a confined SW into a wide film [4,5,6]. In the present work, we tackle theoretically the emission of CSWBs in full, soft thin films; the latter could be stimulated by the tip of a microstrip antenna. We restrict ourselves to applied fields below ferromagnetic resonance (FMR). After dimensionless rewriting of the DR, only three free parameters are left: the ratio of dipolar-exchange length to film thickness η=lex/d, the normalized SW frequency ν=ω/(γ*Js) (Js: saturation induction), and the reduced applied field h=Ba/Js. We then investigate the properties of caustic points numerically and analytically, using suitable approximations. An experimental situation corresponds to fixed film thickness; we therefore record the caustic wavenumbers (WNs) kc*d, beam directions θvc and wavefront directions φc as a function of ν and h, at fixed η. As the example in Fig.2 shows, caustic points can exhibit a rich behaviour. Starting with the caustic WNs, one can see a low-frequency, low-field pocket in the dipolar-dominated regime. Above its sharp boundary, a relatively wide band appears, corresponding to kc*d varying slowly in the dipolar-exchange regime. A final transition brings kc*d into the exchange-dominated regime; the latter has an upper boundary above which no caustic point exists (blue region in Fig.2). The map of φc(h,ν) displays a similar aspect. In the bottom pocket, the caustic wavefront angle is almost constant, with a numerically found limit of about 54.7°. Our analytical approximation yields a value of arccos(1/√3)≈54.7°, in excellent agreement. The corresponding limit for θvc is numerically 109° and 103° from analytics. In the dipolar-exchange band, by contrast with kc*d, φc varies significantly and saturates at 90° in the exchange-dominated region. This saturation is in stark correlation to that of the caustic beam direction θvc, which also goes to 90°. All the above properties can be understood from the geometrical evolution of the DR curve when ν increases, as illustrated in Fig.1. The markers on the curves correspond to their caustic point (2.5 GHz omitted for clarity). At the bottom pocket's boundary, the tangent to the DR at kc*d passes close to the origin: hence, ∂kc/∂φc is large. In the exchange-dominated regime, the curve flattens near φ=π/2. This accelerates the increase of kc*d (cf. curves at 16 and 18.5 GHz) while bringing θvc and φc to π/2 at a certain threshold frequency. Beyond the latter, the curve rounds more and more: θv can no longer be stationary, thus, no caustic point exist. From this investigation, we can thus predict two very different types of CSWBs. At the low-kc pocket's upper boundary, due to the very straight and oblique DR curve, the CSWB should feature a broad wavelength spectrum. By contrast, at the upper boundary of the caustic band, the CSWB's wavelength spectrum should be much narrower. Indeed, the DR at kc*d is tangent to a circle k*d=cst. In addition, since this CSWB consists of two beams with overlapping directions, its amplitude should be correspondingly larger. This phenomenon has been studied theoretically by Kim et al. [7]. The first type of CSWB should be invisible in magneto-optical microscopy, but Brillouin light scattering should reveal the broadband SW beam. The second type of CSWB would require time-resolved Scanning Transmission X-ray Microscopy due to the higher kc.

**References:**

[1] V. Sluka, T. Schneider, R. A. Gallardo et al., Nature Nanotechnology, Vol.14, 4, pp.328-333 (2019) [2] Y. A. Kravtsov and Y. I. Orlov, "Caustic, Catastrophes and Wave Fields", Springer-Verlag 1993 [3] B. A. Kalinikos and A. N. Slavin, Journal of Physics C: Solid State Physics, Vol.19, 7013, pp.7013-7033 (1986) [4] V. E. Demidov, S. O. Demokritov, D. Birt et al., Physical Review B, Vol.80, 014429 (2009) [5] T. Schneider, A. A. Serga, A. V. Chumak et al., Physical Review Letters, Vol.104, 197203 (2010) [6] F. Heussner, A. A. Serga, T. Brächer et al., Applied Physics Letters, Vol.111, 122401 (2017) [7] J.-V. Kim, R. L. Stamps and R. E. Camley, Physical Review Letters, Vol.117, 197204 (2016)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/015b984df791782c0abb0d98462393a9.jpg)
Fig.1: Polar plot of the normalized SW wave number k*d (d: film thickness) as a function of the angle φ from the applied field to the wave vector. The DRs are calculated from the model by Kalinikos and Slavin [3], for an applied induction Ba of 20 mT, frequencies ω/(2π) of 2.5, 8.5, 11, 13.5, 16 and 18.5 GHz, a reduced gyromagnetic ratio γ/(2π)=29.3 GHz/T, saturation induction Js=μ0*Ms=0.92 T, film thickness 60 nm and dipolar-exchange length lex=6.2 nm. Owing to the DR's symmetries, the curves shown here generates the whole polar plots by line symmetry around the quadrant boundaries.

![](https://assets.underline.io/uploads/markdown_image/1/image/80218c582c1cedf81dd66f75f10ca21a.jpg)
Fig.2: Gray-level map of the normalized caustic wave number kc*d as a function of h and v, for η=0.10355. The yellow curve corresponds to FMR; the red filled region below is not considered here. In the top left corner's blue filled region, no caustic points exist.

 Investigation of Caustic Spin Wave Beams in Soft Thin Films

Bose-Einstein condensate is a collective phenomenon which takes place when many bosons – particles having integer value of spin – experience transition to the same single quantum state. The famous example is formation of Cooper pairs of electrons, responsible for the superconducting properties of some materials at extremely low temperatures. Magnons – quasi-particles corresponding to spin waves propagating in magnetic media – are bosons as well. Hence, if we describe oscillation of magnetization as a set of magnons, the phenomenon like Bose-Einstein condensation should also exist in this case. Indeed, the Bose-Einstein condensation of magnons at room temperature has been observed under parametric pumping in thick ferrite films [1], and, recently, by rapid cooling in ultra- thin films [2]. The stability of Bose-Einstein condensate of magnons (mBEC) is determined by the nonlinear interaction between magnons. This includes both interaction between magnons in the same spectral minimum (self-interaction), and the interaction between magnons belonging to two different spectral minima (cross-interaction). mBEC can be stable if the coefficient of nonlinear interaction is positive, i.e. if the interaction is repulsive [3]. It is well known that the self-interaction between magnons in relatively thick films is attractive [3]. However, mBEC has been experimentally proven to be stable [1, 2, 4]. It was shown in [5] that coefficient of nonlinear interaction in mBEC is negative, but rather small, so relatively strong repulsive interaction between magnons in different spectral minima may provide stability to mBEC. Recently, it has been also shown [6] that it is possible to separate mBEC in two parts, each containing magnons from only one spectral minimum, which means that mBEC in each of the minima is stable, and this stability is independent of the strong cross-interaction of magnons from different minima. Thus, the theory developed in [5] cannot be used to explain stability of separated mBEC. Also, the theory [5] assumes a uniform profile of a magnon mode across the film thickness, and infinite in-plane sizes of the magnetic film sample. Both these simplifications are, generally, incorrect. Due to the long-range nature of the dipolar interaction, finite-size effects may have a significant impact on the nonlinear interactions of magnons [4]. Here, we present a simple, but rather accurate model of the nonlinear interactions in mBEC that takes into account the finite-size effects. Using Landau-Lifshits equation and vector Hamiltonian approach [7], we derived approximate analytical expressions for the linear dispersion characteristics and nonlinear interaction coefficients of magnons near the minima of the magnon spectrum (i.e. in the mBEC region). We showed that magnons undergoing mBEC can be described as a gas of weakly-interacting bosonic particles in an infinite potential well formed by a ferromagnetic film. The rest energy of the magnons in mBEC is determined by the dipolar interaction inside a film sample. This leads to a strong anisotropy of the effective mass of magnons in the magnetic film. In the direction along the applied in-plane magnetic field, the effective mass is m∥ = hMs/(32πγA) (where Ms is the saturation magnetization, h is the Planck constant, γ is the gyromagnetic ratio, and A is the exchange stiffness). For yttrium-iron garnet (YIG), this “longitudinal” mass is m∥ ≈ 1.5×10-30 kg ≈ 1.6 me (me is the electron mass). In the directions perpendicular to the applied field magnons are described by the “perpendicular” mass m⊥ = (18λ/d)m∥, which depends on the ratio of the exchange length λ = (2A/μ0Ms2)1/2 to the thickness of the film d. For typical experimental conditions m⊥ « m∥; for example, m⊥ ≈ 0.06 m∥ in a YIG film of the thickness d = 5 μm. The huge anisotropy of the effective mass leads to the anisotropy of the dynamical mBEC properties. In particular, the speed of the Bogolyubov waves [8] depends strongly on the direction of the wave propagation. Since the perpendicular mass m⊥ depends on the thickness of the film, it is possible to achieve the inverse mass relation m⊥ &gt; m∥ in ultra-thin YIG films with d &lt; 300 nm. Under the assumption of totally pinned spins, which is an accurate approximation in the mBEC region, we derived approximate analytical expressions for the nonlinear frequency shift due to the self- and cross- interactions in mBEC: Δf = (Fself + 2Fcross)n; Fself = F0 (2Nd – 3πλ/d√2); Fcross = F0. Here F0 = γ2μ0h/4π2 is the normalisation constant, n is the magnon density, Nd is the component of the demagnetization tensor N = diag(Nx, Ny, Nz) corresponding to the direction along the applied in-plane magnetic field. These expressions show that the nonlinear cross-interaction stays large and positive independently of the confinement of the sample in the applied field direction, while the self-interaction coefficient changes its sign when Nd become greater than the critical value Ncr: Ncr = (3π/2√2)(λ/d). The result explains possibility to separate mBEC into two stable components corresponding to the different minima of the mBEC spectrum [4, 6]. The developed model allows one to make simple and accurate predictions about the stability of the mBEC depending on the shape of the sample used, and to describe dynamics of fluctuations and waves of magnon density in the mBEC. Also, it makes possible to directly apply theoretical methods developed to describe condensates of atoms to study the mBEC dynamics, and, therefore, it might be very useful in further investigations of coherent magnonic phenomena.

**References:**

[1] S. O. Demokritov, V. E. Demidov, O. Dzyapko et al., Nature, vol. 443, pp. 430-433, 2006. [2] M. Schneider, T. Brächer, D. Breitbach et al., Nat. Nanotechnol., vol. 15, pp. 457–461, 2020. [3] V. S. L’vov, Wave turbulence under parametric excitation, Springer Series in Nonlinear Dynamics (Springer, Berlin, 1994). [4] I. V. Borisenko, B. Divinskiy, V. E. Demidov et al., Nat. Commun., vol. 11, no. 1691, 2020. [5] O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk et al., Phys. Rev. B, vol. 96, no. 064438, 2017. [6] I. V. Borisenko, V. E. Demidov, V. L. Pokrovsky et al., Sci. Rep., vol. 10, no. 14881, 2020. [7] V. Tyberkevych, A. Slavin, P. Artemchuk, et al., arXiv:2011.13562 (2020). [8] D. A. Bozhko, A. J. E. Kreil, H. Yu. Musiienko-Shmarova et al., Nat. Commun., vol 10, no. 2460, 2019.

**Images:**

 A Model for Description of Linear Properties and Stability of Bose-Einstein Condensate of Magnons at Room Temperature

The magnetism in curved geometries is attracting attention because of several theoretical predictions with possible potential applications, such as a curvature induced effective Dzyaloshinskii-Moriya like interaction [1]. Within this setting, analogously to a geometric curvature, a curvature induced by a gauge field [2] is a fruitful aspect to consider in the interaction of the magnetization with an applied static electric field. In a quantum description of the motion of a magnetic moment, an electric field does not modify the energy but it rather modifies the linear momentum, therefore introducing an additional phase for the corresponding Schrödinger wave function, an effect which is known as the Aharonov-Casher’s effect [3]. The classical analog is the spin wave of a ferromagnet. It has been shown that a spin wave, which is carrying a magnetic moment, will acquire a phase when the system is subjected to an electric field [4]. This effect, even if rather small, can be experimentally observed [5]. It is therefore of interest to extend this idea to a general micromagnetic description of the problem. The aim of this research is to pose the problem of the micromagnetics under the effect of an electric field by using the so-called gauge invariance. Briefly the spin-waves, which are small oscillations, deviate from the uniform magnetization with respect to a local reference frame and around the ez direction. They interact with an applied electric field E Fig.(1) modifying the spin wave dispersion relation. Therefore, according to the so-called local exchange invariance [6], we propose a phase shift model, in the exchange spin waves regime, in order to possibly describe and interpret the data shown in [5]. More in details the local gauge symmetry is approximately a local invariance property (because of its thermodynamic origin), of the lagrangian density of the magnetization m(x, t) of a ferromagnet. It is known that it is invariant with respect to the SO(3) global symmetry [7]. In our case this symmetry reduces to a unitary representation of the SO(2) group, (i.e., U(1)). Due to the reason that this symmetry has to be local (i.e., to be position dependent) the invariance of the lagrangian density is broken unless introducing a gauge field able to solve the technical problem of the parallel transport of the magnetization vector. Therefore we show that a particular orientation of a reference frame has no physical relevance. In fact, gauge invariance should occur whenever one can define non-collinear, local reference frames. In our model, the orientation of the local non-coordinate reference frames cannot be uniquely defined because of the presence of an electric field. Summarizing in our invariant gauge formulation the reference frames’ orientation is unspecified with respect to a local phase which defines an invariance property of the exchange interaction terms in the lagrangian density of the magnetization m(x, t). The Lagrangian density of a ferromagnet, interacting with an electric field, is transformed into a new Lagrangian density upon a formal redefinition of the standard partial derivative. Moreover the transformation group, which expresses this invariance, imposes definite restrictions on the dispersion relation that at the first order in a Tylor expansion was previously found to be ω=ω(ky)-γLc-2∂ω/∂kyEx [4], with γL the gyromagnetic ratio and c the speed of light. Therefore our findings point to a more advanced theoretical interpretation with respect to previous discoveries [4] in order to drive experiments in manipulating spin waves and developing electrically tunable magnonic devices.

**References:**

[1] R. Streubel et al., J. Phys. D: Appl. Phys. 49, 363001 (2016). [2] S.G. Tan, et al., Physics Reports, Volume 882, Pages 1-36, (2020). [3] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319, (1984). [4] V. Basso and P. Ansalone, EPL, Vol. 130, Number 1, Article 17008, (2020). [5] X. Zhang, T.Liu, M.E. Flatté and H.X. Tang Phys. Rev. Lett. 113, (2014). [6] I.E. Dzyaloshinskii and G.E.Volovik, J.Physique, 39, 693, (1978). [7] W. F. Brown, Magnetoelastic interactions, pp. 26 and 79, (Vol. 9), Berlin: Springer, (1966).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/62bbb322a9e11ec137f86fd168238d14.jpg)
Fig. 1 The system under study is a two dimensional infinite ferromagnetic layer with primary magnetization along the direction ez , an external electric field is applied perpendicularly to both the magnetization and k spin-wave vector.

 Effect of the Local Exchange Invariance on the Magnetization Dynamics in a Ferromagnet

wo recent reports of the Münster University experimental team led by S.O. Demokritov [Borisenko et al., Nat. Comm. 2020, Borisenko et al., Sci. Rep. 10, 14881 (2020)] displayed several important facts contradicting to the existing theories of the Bose-Einstein condensate of magnons (BECM) [Tupitsyn et al. 2008, Rezende 2009, F. Li et al. 2013]. We present simplified theoretical arguments to determine what are properties of the uniform stationary BECM established under the action of permanent parametric pumping that follows from the experimental facts. We point out that the complete equilibrium was not established in the magnon system because the inter-condensates relaxation time is much longer than the lifetime of condensate magnons. Since the magnons repulse each other, the uniform stationary state of the condensate does not collapse. If the inter-condensates relaxation processes are negligible, then the stationary state of the condensate must be symmetric since the processes of parametric pumping and those responsible for finite magnons lifetime are identical for the two condensates. The weak inter- condensates relaxation processes lead to a weak spontaneous violation of the use of reflection symmetry. This prediction agrees with the inter-condensate interference structure of the condensate discovered by Novik-Boltyk et al. [Novik- Boltyk 2012]

**References:**

 Borisenko, I., Divinskiy, B., Demidov, V. et al. Direct evidence of spatial stability of Bose-Einstein condensate of magnons. Nat Commun 11, 1691 (2020). Borisenko, I.V., Demidov, V.E., Pokrovsky, V.L. et al. Spatial separation of degenerate components of magnon Bose–Einstein condensate by using a local acceleration potential. Sci Rep 10, 14881 (2020) Tupitsyn, I. S., Stamp, P. C. E. & Burin, A. L. Stability of Bose–Einstein condensates of hot magnons in yttrium iron garnet films. Phys. Rev. Lett. 100, 257202 (2008). Rezende, S. M. Theory of coherence in Bose–Einstein condensation phenomena in a microwave-driven interacting magnon gas. Phys. Rev. B 79, 174411 (2009) Li, F., Saslow, W. M. & Pokrovsky, V. L. Phase diagram for magnon condensate in yttrium iron garnet film. Sci. Rep. 3, 1372 (2013) Nowik-Boltyk, P., Dzyapko, O., Demidov, V. E., Berloff, N. G. & Demokritov, S. O. Spatially non-uniform ground state and quantized vortices in a two-component Bose–Einstein condensate of magnons. Sci. Rep. 2, 482 (2012)

**Images:**

 Stationary state of Bose-Einstein condensate of magnons: theory and experiment

Due to low transport dissipation property, magnon was considered as a candidate of next-generation carrier of spintronic industry. In 2016, the van Wees group[1] found an interesting difference between electrically and thermally excited magnons on the nonlocal transport in a very short range. Unlike electrically excited magnons, thermally excited magnons showed a sign-reversal in less than about 300 nm distance between heat injector and magnons detector. Some people developed a bulk SSE theory to explain this phenomenon. According to this theory, a heat flow Jq in YIG will excite a thermal magnon flow Jm,q along with it, A diffusive magnon flow is induced to balance the thermal magnon flow, until the system reaches a steady-state. The sign-reversal derived from a competition of these two magnon flow. While in our point of view, this theory may not reach the nature of transmission properties of thermally excited magnons. To clarify the physical essence and gain complete information in the reciprocal space of this process, we adapted mesoscopic Boltzmann method to study magnon transports[2]. In the method, like phonon, we introduced a special scattering process named “Normal Scattering”, which prevents magnons’ momenta during scattering process. When normal scattering is dominant, magnons turned to a uniform velocity. This uniform velocity makes magnons flux acting as a viscous fluid. Based on Boltzmann method, we developed a new diffusion method of magnons viscous fluid to simplify the calculation and found its nonlinear property. Two-dimensional figures of magnon flux showed the viscosity of magnon fluid created two eddies near the heat source. And the numerical result could fit Experimental data precisely. Our results show magnon flow may be controlled like a viscous fluid by applied magnetic field, which has promising applications in nanoscale magnon-based spintronic devices.

**References:**

1, J. Shan, L. Cornelissen, N. Vlietstra, J. Youssef, T. Kuschel, R. Duine, and B. van Wees, Phys. Rev. B 94, 174437 (2016) 2, T. Liu, W. Wang & J. Zhang, Phys. Rev. B, 99, 214407 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6442af17d25c4eca7d4827e3e6b415bc.jpg)

 Novel Transport Properties of Viscous-Fluid type Magnon in Magnetic Thin Film

Studying spin system dynamics concerns different rapidly growing fields of technology such as developing new types of information transfer systems [1], quantum computing [2, 3], new approaches for magnetic recording and reading [4]. The most common way to excite spin oscillations is by using the inverse Faraday effect (IFE). It is based on a Raman-like coherent optical scattering process and does not require absorption of light. This gives us two main advantages of this effect: it is instantaneous and non-thermal [5]. IFE results in magnetization induced by high-power laser radiation according to the following formula: Min=χ/16π [E×E*], (1) where χ is magneto-optical susceptibility, E is the electric field of the incident light, and E* is its complex conjugate. The cross product in formula (1) is zero for linearly polarized incident light and is maximum for circularly polarized light. The excitation of high order spin waves requires the nonuniform distribution of the electromagnetic field in the medium. It is commonly achieved by using the nanostructured materials in which the resonant phenomena can occur breaking the uniformity of the field. For example, in work [6] the all-dielectric 1D grating was used to generate spin waves due to the excitation of propagating waveguide modes. Another way to excite spin waves is the utilization of anisotropic materials in which the polarization conversion of the incident light occurs. Such a transformation leads to the spatial dependency of magnetization induced via IFE. This nonuniformity of the magnetic torque allows one to excite the spin waves of a high order. Let us consider the case of biaxial crystals the influence on the incident light polarization of which is described by the permittivity tensor εij. The diagonal components of this tensor are assumed not to be equal ε11≠ε22≠ε33 and connected with cartesian coordinates x, y, and z respectively. Let the light fall under the normal incidence to the plane of the material and propagate along the z-axis. Placing the material under study in an external magnetic field along the z-axis leads to the appearance of additional off-diagonal components which are ε12=-i×g and ε21=i×g. Polarization of the light propagating in such a medium transforms due to the anisotropy and Faraday effect. This process is described theoretically in [7]. To be more specific, consider the YFeO3 crystal with an orthorhombic cell. Its crystallographic axes a, b, and c are oriented along x, y, and z axes respectively. The main refractive indices of such material are n1=2.365, n2=2.4, n3=2.337 at the wavelength of 633 nm (ε'' is assumed to be zero). The transformation of the incident light with circular polarization when it passes through anisotropic medium results in oscillations of the magnetization induced via IFE (blue line on Fig. 1). It worth noting that oscillations of magnetization occur for linearly polarized light. The maximum value of cross-product in formula (1) can be achieved for the polarization of incident light at an angle of 45° to the a crystal axis (green line on Fig. 1). In this case, the material behaves as a quarter-wave plate fully transforming the initial linear polarization to circular. One can observe the oscillations even for light polarized linearly along the crystal axis but their maximum value is two times smaller compared to the previous scenarios (Fig. 2). In this situation, the polarization transformation due to birefringence must disappear but the Faraday effect slightly rotates the polarization off the axis allowing the anisotropy to take its place. The oscillations of IFE due to the birefringent properties of the material under study have been theoretically demonstrated. Such a behavior of the induced magnetization is important for generating spin waves of a high order allowing one to avoid additional nanostructuring of the material. This work was supported by the Ministry of Science and Higher Education of the Russian Federation (mega-grant no. 075-15-2019-1934).

**References:**

[1] S. Neusser and D. Grundler, Advanced Materials, Vol. 21 p. 2927 (2009) [2] K. Ganzhorn, S. Klingler, T. Wimmer, et al, Applied Physics Letters, Vol. 109, p. 022405 (2016) [3] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, et al, Applied Physics Express, Vol. 12, p. 070101 (2019) [4] D.O. Ignatyeva, C.S. Davies, D.A. Sylgachev, et al, Nature Communications, Vol. 10, p. 1 (2019) [5] A.V. Kimel, A. Kirilyuk, P.A. Usachev, et al, Nature, Vol. 435, p. 655 (2005) [6] A.I. Chernov, M.A. Kozhaev, D.O. Ignatyeva, et al, Nano Letters, Vol. 20, p. 5259 (2020) [7] A.K. Zvezdin, V.A. Kotov, Modern magnetooptics and magnetooptical materials, CRC Press (1997)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9fd7730e609d00e4f4b1fe8becda2c13.jpg)
Fig. 1. Cross product of electric field and its complex conjugate depending on the propagation length for circularly (blue line) and linearly polarized at an angle of 45° to the crystal axis (green line) incident light.

![](https://assets.underline.io/uploads/markdown_image/1/image/a63c9245721032602764f78c072ba99b.jpg)
Fig. 2. Cross product of electric field and its complex conjugate depending on the propagation length for incident light polarized along the a crystal axis.

 Oscillating Behavior of Inverse Faraday Effect in YFeO3

Cavity magnonics built on strongly coupled cavity photons and magnons has gradually demonstrated its unique advantages in fundamental and applied research in the past few years. It has several advantages, including flexible turnability, excellent controllability, and high compatibility. Furthermore, a versatile platform based on coherently coupled microwave photons, magnons, phonons, superconducting qubits, and optical photons has begun to take shape. The hybrid system is expected to be applied in quantum information processing, transfer, and storage. Before 2018, all experiments were carried out in the case of coherent coupling between cavity photons and magnons. In 2018, an intriguing dissipative coupling between cavity photon and magnon was observed for the first time, which originated from the cooperative dissipation of the cavity mode and magnon mode. Unlike the coherent coupling case, where the spectrum is shown as an anti-crossing between cavity mode and magnon mode, the spectrum is shown as a level attraction in the case of dissipative coupling. The revelation of dissipative coupling greatly enhances the richness and functionality of the cavity magnonics and stimulates some fundamental research. We will introduce this trend with two recent works from our group. After revealing the dissipative coupling mechanism, we first realized nonreciprocity and unidirectional invisibility in cavity magnonics by utilizing the interference effect between the coherent and dissipative couplings. The isolation ratio can be relatively large with optimized small insertion loss. With the dissipative coupling mechanism and magnon mode linewidth engineering technique, we construct an anti-parity-time (APT) symmetric cavity magnonics system. Besides the dissipative coupling, the APT symmetric system also needs (i) the cavity mode and magnon mode have the opposite frequency, which can be effectively realized with a detuned cavity magnonics system in the rotating frame, (ii) the same damping or gain rates of the cavity mode and magnon mode, which can be engineered by adjusting the damping rate of the magnon mode to match that of the cavity mode. Further, we observed the exceptional points in this APT symmetric cavity magnonics system, which are the transition points between the PT-symmetry-preserved and PT-symmetry-broken phases. Between two EPs, the maximal coherent superposition of photon and magnon states is robustly sustained by the preserved APT symmetry. Moreover, we observed another kind of singularity, arising from the dissipative coupling of two antiresonances, is an unconventional bound state in the continuum (BIC). At the settings of BICs, the coupled system exhibits infinite discontinuities in the group delay. We find that both singularities coexist at the equator of the eigenvector mapped Bloch sphere, revealing a unique hybrid state that simultaneously exhibits the maximal coherent superposition and slow light capability. We expect to explore more interesting physics and applications based on the dissipative coupling in cavity magnonics.

**References:**

1. H. Huebl, et al, Phys. Rev. Lett. 111, 127003 (2013). 2. Y. Tabuchi, wt al, Phys. Rev. Lett. 113, 083603 (2014). 3. X. Zhang, et al, Phys. Rev.Lett. 113, 156401 (2014). 4. M. Goryachev, et al, Phys. Rev. Applied 2, 054002 (2014). 5. L. Bai, et al, Phys. Rev. Lett. 114, 227201 (2015). 6. Y.-P. Wang, Phys. Rev. Lett. 123, 127202 (2019). 7. Y. Yang, Phys. Rev. Lett. 125, 147202 (2020).

**Images:**

 Nonreciprocity and Unconventional Singularities in Cavity Magnonics INVITED

The field of magnonics considers the operations with data carried by spin waves and their quanta, magnons [1]. Traditionally, magnonics has been a room-temperature research discipline. Nowadays, the new direction of cryogenic magnonics attracts increasing attention. The decrease in temperature, in particular, results in the reduction of thermal population n of magnons below 1 opening access to single-magnon quantum physics [2]. Moreover, cryogenic temperatures allow for combining magnonics with superconductivity [3-5]. Ferromagnetism and superconductivity are fundamental cooperative phenomena in condensed matter physics. Entailing opposite spin orders, they share an essential conceptual similarity: Disturbances in magnetic ordering in magnetic materials can propagate in the form of spin waves (magnons) while magnetic fields penetrate the majority of technologically relevant (type II) superconductors as a lattice of magnetic flux quanta (fluxons, Abrikosov vortices). Despite a rich choice of wave and quantum phenomena predicted, magnon-fluxon coupling has lacked experimental scrutiny so far. In this talk, a selection of our recent results [5] on the interaction of spin waves with the flux lattice in ferromagnet/superconductor Py/Nb bilayers will be presented - see Fig. 1. In this system we found the magnon frequency spectrum to exhibit a Bloch-like band structure (as shown in Fig. 1(b)) which can be tuned by the biasing magnetic field. Furthermore, the frequency spectra of spin waves scattered on the flux lattice moving under the action of a transport current in the superconductor exhibit Doppler shifts. The resonance absorption of spin waves is explained by their inelastic scattering on the vortex lattice constituting a reconfigurable magnonic crystal [6]. The observed Doppler shifts arise due to the modified dispersion relation for spin waves scattered on the moving vortex lattice constituting a moving Bragg grating. In addition, manipulation of spin waves and tailoring of their transmission spectra in Py will be exemplified by using adjacent Nb layers with nanofabricated pinning potential landscapes. A special attention will be devoted to direct-write hybrid superconductor-based systems with fast relaxation of non-equilibrium effects and supporting ultra-fast vortex motion at &gt;10km/s velocities [7]. Research leading to these results was done in collaboration with R. Sachser, T. Brächer, T. Böttcher, V. Kruglyak, R. Vovk, V. Shklovskij, M. Huth, B. Hillebrands and A. Chumak.

**References:**

[1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, et al., Nat. Phys. 11 (2015) 453 [2] D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi,et al., Science 367 (2020) 425 [3] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, et al., Adv. Func. Mater. 28 (2018) 1802375 [4] A. A. Bespalov, A. S. Mel’nikov, and A.I. Buzdin, Phys. Rev. B 89 (2014) 054516 [5] O. V. Dobrovolskiy, R. Sachser, T. Brächer, et al., Nat. Phys. 15 (2019) 477 [6] A. V. Chumak, A. A. Serga, and B. Hillebrands, J. Phys. D: Appl. Phys. 50 (2017) 244001 [7] O. V. Dobrovolskiy, D. Yu. Vodolazov, F. Porrati, et al., Nat. Commun. 11 (2020) 3291

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/02511907edbb80ebd00a971a77b36b08.jpg)
(a) Sketch of the experimental system. (b) Fluxon-induced reconfigurable magnonic crystal: Bandgaps in the spin-wave transmission spectra in Py are observed when the Nb layer contains an array of Abrikosov vortices.

 Magnon Fluxonics INVITED

Acoustic wave has been widely integrated into modern electronics to enable various functions, including sensing, communication, signal filtering, etc. Conventionally, acoustic waves propagate equally forward and backward. Here, we employ a fundamentally new mechanism, magneto-rotation coupling[1], to break the symmetry of the propagation, and consequently induces an unprecedented rectification ratio to 100% at the optimized condition, in accordance with the theoretical prediction. As a unique feature of surface acoustic waves, individual lattice point undergoes a rotation motion, which changes its sign according to the wave propagation direction (see the blue and red oriented circles in figure 1(a)). More than 40 years ago, Maekawa and Tachiki developed a theoretical framework [1], proposing the coupling between the lattice rotation and the magnetization via magnetic anisotropy, namely magneto- rotation coupling. Thus, to break the symmetry of the propagation, here, we employ magneto-rotation coupling to capture the chirality of the acoustic waves in a perpendicularly anisotropic ultra-thin film Ta/CoFeB(1.6 nm)/MgO (see figure 1(b)), achieving a giant nonreciprocal acoustic wave attenuation (see figure 1(c)) with a rectification ratio up to 77% [2].

**References:**

[1] S. Maekawa and M. Tachiki, Surface acoustic attenuation due to surface spin wave in ferro and antiferromagnets, AIP Conf. Proc. 29, 542 (1976). [2] M. Xu, K. Yamamoto, J. Puebla, K. Baumgaertl, B. Rana, K. Miura, H. Takahashi, D. Grundler, S. Maekawa, and Y. Otani, Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling, Vol. 6, no. 32, eabb1724 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/404c74cc9b48109c9508f92526b2d977.jpg)
Fig. 1. (a) Schematic of the nonreciprocity mechanism. (b) Illustration of the experimental setup. (c) Attenuation ofacoustic waves, P±k, near a spin-wave resonance condition for surface acoustic wavenumbers +k and -k.

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Evolution of room-temperature magnon gas toward coherent Bose-Einstein condensate

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Evolution of room-temperature magnon gas toward coherent Bose-Einstein condensate

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Spin-wave Circulation in a Ferromagnetic Resonator with Two Adjacent Layers

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